Simple Harmonic Motion Quiz

Simple Harmonic Motion

• Spring oscillator is a simple harmonic vibration.
• The total energy (E) of a spring oscillator is proportional to the product of the mass (m), amplitude (A), and frequency (ω): E = 1/2 mA²ω².

Energy of Simple Harmonic Vibration

• If the mass of a spring oscillator is increased three times and the amplitude is doubled, the total energy becomes 12E.

Synthesis of Simple Harmonic Vibration

• If a particle participates in two or more simple harmonic motions at the same time, its motion is the synthesis of these harmonic motions.
• The natural frequency (ʋ) and natural period (T) of a spring oscillator are related by the equation: ʋ = 2π/T.

Phase and Initial Phase

• The phase (ωt + φ) of a simple harmonic vibration determines the motion state of the vibration system.
• The initial phase (φ) is the phase at time zero, and it is related to the displacement (x) and velocity (V) by the equations: x = Acos(ωt + φ) and V = -Aωsin(ωt + φ).

Trigonometric Functions

• The sine and cosine functions have the following values: sin(0°) = 0, sin(30°) = 1/2, sin(45°) = 1/√2, sin(60°) = √3/2, sin(90°) = 1, cos(0°) = 1, cos(30°) = √3/2, cos(45°) = 1/√2, cos(60°) = 1/2, cos(90°) = 0.

Equation of Simple Harmonic Motion

• The equation of simple harmonic motion is derived from Newton's second law: F = -kx = ma, where F is the elastic force, k is the spring constant, x is the displacement, and a is the acceleration.
• The acceleration of a simple harmonic motion is given by the equation: a = -ω²x, where ω is the angular frequency.
• The solution to the motion-differential equation is: x = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the initial phase.